Quantum quenches in integrable field theories

Abstract

We study the non equilibrium time evolution of an integrable field theory in 1 + 1 dimensions after a sudden variation of a global parameter of the Hamiltonian. For a class of quenches defined in the text, we compute the long times limit of the one point function of a local operator as a series of form factors. Even if some subtleties force us to handle this result with care, there is a strong evidence that for long times the expectation value of any local operator can be described by a generalized Gibbs ensemble with a different effective temperature for each eigenmode.

Dynamic correlations, fluctuation-dissipation relations, and effective temperatures after a quantum quench of the transverse field Ising chain

“…These Ansatze are motivated by the observed rapid temporal decay of these functions, modulated by damped oscillations. [20,60] for the correlation function C x + (t). We found good agreement between the analytical expression (86) and the fit of the numerical data also for the decay of the response function.…”

“…(3), it is necessary to introduce a number of Lagrange multipliers (see further below), one for each "conserved quantity", which eventually turn into a set of "effective temperatures" {T k eff } determined by the condition ψ 0 |n k |ψ 0 = n k T =T k eff . These quantities prove to be particularly useful since they naturally appear in the calculation of (stationary and non-stationary) expectation values [17,18,20]. It was in fact suggested [17] that the stationary behavior of the system after quenches towards Hamiltonians of the form (7) can be described in terms of the density matrixρ GGE obtained by maximizing the von Neumann entropy S[ρ] under the constraints on the expectation values of n k .…”

“…Note that Eqs. (19) and (20) are not equivalent, unless β * AB eff is allowed to depend on the frequency. In addition, while for given R AB (t) andC AB + (ω) it is always possible to define the frequency-dependent inverse temperature β AB eff (ω) from Eq.…”

“…(69) contribute to the leading order of Eq. (20), T * eff can be interpreted as a temperature associated with the oscillatory frequency ω = 8, corresponding to the threshold value ω max . Consistently with this interpretation, the effective temperature T M eff (ω) defined on the basis of Eq.…”

“…Integrable systems, instead, are not expected to thermalize. However, their asymptotic stationary state should nonetheless be described by the so-called generalized Gibbs ensemble (GGE) in which each conserved quantity is characterized by a generally different effective temperature [17][18][19][20][21][22][23]. On the other hand, other works [24][25][26][27][28][29][30][31][32][33][34][35] have shown (or at least argued) that this scenario could actually be significantly richer.…”